We know that when we count we start 1,2 …. . But there are other numbers

like 0, negative numbers and decimals. All these types of numbers are

categorized in different groups like counting numbers, integers,real

numbers, whole numbers and rational and irrational numbers according to

their properties. all this have been covered in this chapter

categorized in different groups like counting numbers, integers,real

numbers, whole numbers and rational and irrational numbers according to

their properties. all this have been covered in this chapter

Integers

Identify integers

Consider a number line below

The

numbers from 0 to the right are called positive numbers and the numbers

from 0 to the left with minus (-) sign are called negative numbers.

Therefore all numbers with positive (+) or negative (-) sign are called

integers and they are denoted by Ζ. Numbers with positive sign are

written without showing the positive sign. For example +1, +2, +3, …

they are written simply as 1, 2, 3, … . But negative numbers must carry

negative sign (-). Therefore integers are all positive and negative

numbers including zero (0). Zero is neither positive nor negative

number. It is neutral.

numbers from 0 to the right are called positive numbers and the numbers

from 0 to the left with minus (-) sign are called negative numbers.

Therefore all numbers with positive (+) or negative (-) sign are called

integers and they are denoted by Ζ. Numbers with positive sign are

written without showing the positive sign. For example +1, +2, +3, …

they are written simply as 1, 2, 3, … . But negative numbers must carry

negative sign (-). Therefore integers are all positive and negative

numbers including zero (0). Zero is neither positive nor negative

number. It is neutral.

The

numbers from zero to the right increases their values as the increase.

While the numbers from zero to the left decrease their values as they

increase. Consider a number line below.

numbers from zero to the right increases their values as the increase.

While the numbers from zero to the left decrease their values as they

increase. Consider a number line below.

If

you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater

than 2. We use the symbol ‘>’ to show that the number is greater than

i. e. 3 >2(three is greater than two). And since 2 is to the left of

3, we say that 2 is smaller than 3 i.e. 2<3. The symbol ‘<’ is

use to show that the number is less than.

you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater

than 2. We use the symbol ‘>’ to show that the number is greater than

i. e. 3 >2(three is greater than two). And since 2 is to the left of

3, we say that 2 is smaller than 3 i.e. 2<3. The symbol ‘<’ is

use to show that the number is less than.

Consider

numbers to the left of 0. For example if you take -5 and -3. -5 is to

the left of -3, therefore -5 is smaller than -3. -3 is to the right of

-5, therefore -3 is greater than -5.

numbers to the left of 0. For example if you take -5 and -3. -5 is to

the left of -3, therefore -5 is smaller than -3. -3 is to the right of

-5, therefore -3 is greater than -5.

Generally, the number which is to the right of the other number is greater than the number which is to the left of it.

If

two numbers are not equal to each to each other, we use the symbol ‘≠’

to show that the two numbers are not equal. The not equal to ‘≠’ is the

opposite of is equal to ‘=’.

two numbers are not equal to each to each other, we use the symbol ‘≠’

to show that the two numbers are not equal. The not equal to ‘≠’ is the

opposite of is equal to ‘=’.

Example 21

Represent the following integers Ζ on a number line

- 0 is greater than Ζ and Ζ is greater than -4
- -2 is less than Ζ and Ζ is less than or equal to 1.

Solution

a.

0 is greater than Ζ means the integers to the left of zero and Ζ is

greater than -4 means integers to the left of -4. These numbers are -1,

-2 and -3. Consider number line below

0 is greater than Ζ means the integers to the left of zero and Ζ is

greater than -4 means integers to the left of -4. These numbers are -1,

-2 and -3. Consider number line below

b.

-2 is less than Ζ means integers to the right of -2 and Ζ is less than

or equal to 1 means integers to the left of 1 including 1. These

integers are -1, 0 and 1. Consider the number line below

-2 is less than Ζ means integers to the right of -2 and Ζ is less than

or equal to 1 means integers to the left of 1 including 1. These

integers are -1, 0 and 1. Consider the number line below

Example 22

Put the signs ‘is greater than’ (>), ‘is less than’ (<), ‘is equal to’ (=) to make a true statement.

Addition of Integers

Add integers

Example 23

2 + 3

Show a picture of 2 and 3 on a number line.

When

drawing integers on a number line, the arrows for the positive numbers

goes to the right while the arrows for the negative numbers goes to the

left. Consider an illustration bellow.

drawing integers on a number line, the arrows for the positive numbers

goes to the right while the arrows for the negative numbers goes to the

left. Consider an illustration bellow.

The

distance from 0 to 3 is the same as the distance from 0 to -3, only the

directions of their arrows differ. The arrow for positive 3 goes to the

right while the arrow for the negative 3 goes to the left.

distance from 0 to 3 is the same as the distance from 0 to -3, only the

directions of their arrows differ. The arrow for positive 3 goes to the

right while the arrow for the negative 3 goes to the left.

Example 24

-3 + 6

Solution

Subtraction of Integers

Subtract integers

Since

subtraction is the opposite of addition, if for example you are given

5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can

use a number line in the same way as we did in addition. Therefore 5-4

on a number line will be:

subtraction is the opposite of addition, if for example you are given

5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can

use a number line in the same way as we did in addition. Therefore 5-4

on a number line will be:

Take five steps from 0 to the right and then four steps to the left from 5. The result is 1.

Multiplication of Integers

Multiply integers

Example 25

2×6 is the same as add 2 six times i.e. 2×6 = 2 + 2 + 2 + 2 + 2 +2 = 12. On a number line will be:

Multiplication

of a negative integer by a negative integer cannot be shown on a number

line but the product of these two negative integers is a positive

integer.

of a negative integer by a negative integer cannot be shown on a number

line but the product of these two negative integers is a positive

integer.

From

the above examples we note that multiplication of two positive integers

is a positive integer. And multiplication of a positive integer by a

negative integer is a negative integer. In summary:

the above examples we note that multiplication of two positive integers

is a positive integer. And multiplication of a positive integer by a

negative integer is a negative integer. In summary:

- (+)×,(+) = (+)
- (-)×,(-) = (+)
- (+)×,(-) = (-)
- (-)×,(+) = (-)

Division of Integers

Divide integers

Example 26

6÷3 is the same as saying that, which number when you multiply it by 3 you will get 6, that number is 2, so, 6÷3 = 2.

division is the opposite of multiplication. From our example 2×3 = 6

and 6÷3 = 2. Thus multiplication and division are opposite to each

other.

Dividing

two integers which are both positive the quotient (answer) is a

positive integer. If they are both negative also the quotient is

positive. If one of the integer is positive and the other is negative

then the quotient is negative. In summary:

two integers which are both positive the quotient (answer) is a

positive integer. If they are both negative also the quotient is

positive. If one of the integer is positive and the other is negative

then the quotient is negative. In summary:

- (+)÷(+) = (+)
- (-)÷(-) = (+)
- (+)÷(-) = (-)
- (-)÷(+) = (-)

Mixed Operations on Integers

Perform mixed operations on integers

You

may be given more than one operation on the same problem. Do

multiplication and division first and then the rest of the signs. If

there are brackets, we first open the brackets and then we do division

followed by multiplication, addition and lastly subtraction. In short we

call it BODMAS. The same as the one we did on operations on whole

numbers.

may be given more than one operation on the same problem. Do

multiplication and division first and then the rest of the signs. If

there are brackets, we first open the brackets and then we do division

followed by multiplication, addition and lastly subtraction. In short we

call it BODMAS. The same as the one we did on operations on whole

numbers.

Example 27

9÷3 + 3×2 -1 =

Solution

9÷3 + 3×2 -1

=3 + 6 -1 (first divide and multiply)

=8 (add and then subtract)

Example 28

(12÷4 -2) + 4 – 7=

Solution

(12÷4 -2) + 4 – 7

=1 + 4 – 7 (do operations inside the brackets and divide first)

=5 – 7 (add)

=2

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